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On the approximability and the selection of particle shape functions

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Summary.

Particle methods, also known as meshless or meshfree methods, have become popular in approximating solutions of partial differential equations, especially in the engineering community. These methods do not employ a mesh, or use it minimally, in the construction of shape functions. There is a wide variety of classes of shape functions that can be used in particle methods. In this paper, we primarily address the issue of selecting a class of shape functions, among this wide variety, that would yield efficient approximation of the unknown solution. We have also made several comments and observations on the order of convergence of the interpolation error, when these shape functions are used; specifically, we have shown that the interpolation error estimate, for certain classes of shape functions, may not indicate the actual order of convergence of the approximation error.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, ACE 6.412, University of Texas at Austin, Austin, TX, 78712

    Ivo Babuška

  2. Department of Mathematics, Syracuse University, 215 Carnegie, Syracuse, NY, 13244

    Uday Banerjee

  3. Department of Mathematics, University of Maryland, College Park, MD, 20742

    John E. Osborn

Authors
  1. Ivo Babuška
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  2. Uday Banerjee
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  3. John E. Osborn
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Corresponding author

Correspondence to Uday Banerjee.

Additional information

Mathematics Subject Classification (2000): 65N15, 65N30, 41A30

This research was supported by NSF Grant # DMS-98-02367 and ONR Grant # N00014-99-1-0724

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Babuška, I., Banerjee, U. & Osborn, J. On the approximability and the selection of particle shape functions. Numer. Math. 96, 601–640 (2004). https://doi.org/10.1007/s00211-003-0489-2

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  • DOI: https://doi.org/10.1007/s00211-003-0489-2

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